Breakdown of Quasilocality in Long-Range Quantum Lattice Models

We study the nonequilibrium dynamics of correlations in quantum lattice models in the presence of long-range interactions decaying asymptotically as a power law. For exponents larger than the lattice dimensionality, a Lieb-Robinson-type bound effectively restricts the spreading of correlations to a causal region, but allows supersonic propagation. We show that this decay is not only sufficient but also necessary. Using tools of quantum metrology, for any exponents smaller than the lattice dimension, we construct Hamiltonians giving rise to quantum channels with capacities not restricted to a causal region. An analytical analysis of long-range Ising models illustrates the disappearance of the causal region and the creation of correlations becoming distance independent. Numerical results obtained using matrix product state methods for the $XXZ$ spin chain reveal the presence of a sound cone for large exponents and supersonic propagation for small ones. In all models we analyzed, the fast spreading of correlations follows a power law, but not the exponential increase of the long-range Lieb-Robinson bound.

Figure 1

Density contour plots of the connected correlator ${⟨{\sigma }_o^x{\sigma }_{\delta }^x⟩}_{\mathrm{c}}$ in the ($\delta$, $t$) plane for long-range Ising chains with $|\mathrm{\Lambda }|=1001$ sites and three different values of $\alpha$. Dark colors indicate small values, and initial correlations at $t=0$ are vanishing.

Figure 2

Top row: Density plots of the correlator ${⟨{\sigma }_o^z{\sigma }_{\delta }^z⟩}_{\mathrm{c}}$ in the ($\delta$, $t$) plane. The results are for long-range $XXZ$ chains with $|\mathrm{\Lambda }|=40$ sites and exponents $\alpha =3/4$, $3/2$, and 3 (from left to right). The left and center plots reveal supersonic spreading of correlations, not bounded by any linear cone, whereas such a cone appears in the right plot for $\alpha =3$. Bottom row: As above, but showing contour plots of $\ln {⟨{\sigma }_o^z{\sigma }_{\delta }^z⟩}_{\mathrm{c}}$ in the ($\ln \delta$, $\ln t$) plane. For better visualization, odd or even effects caused by the staggered initial state have been eliminated (see Supplemental Material [17] for details). All plots in the bottom row are consistent with a power-law-shaped causal region for larger distances $\delta$.